lorentz[x_,t_,v_] = {(x- v t)/Sqrt[1-v^2],(t - v x)/Sqrt[1-v^2]}

Manipulate[
Column[{v,Show[
ListPlot[Flatten[Table[lorentz[x,t,v],{x,-10,10},{t,-10,10}],1], PlotRange->{{-14,14},{-14,14}},AspectRatio->1,ImageSize->Large,PlotStyle->PointSize[Medium], AxesLabel->{x',ct'}],
ListPlot[{lorentz[-80,-100,v],lorentz[80,100,v]},Joined->True,PlotStyle->{Purple}],
ListPlot[{lorentz[-74,-100,v],lorentz[86,100,v]},Joined->True,PlotStyle->{Purple}],
ListPlot[{lorentz[0,-100,v],lorentz[0,100,v]},Joined->True,PlotStyle->{Red}],
ListPlot[{lorentz[8,-100,v],lorentz[8,100,v]},Joined->True,PlotStyle->{Red}],
(*ListPlot[{lorentz[8,2.5,v],lorentz[4,6.5,v]},Joined->True,PlotStyle->{Green}],*)ListPlot[{lorentz[0,0,v],lorentz[6,0,v]},PlotStyle->{PointSize[Large],Red}],
ListPlot[{lorentz[-100,-100,0],lorentz[100,100,0],lorentz[100,-100,0],lorentz[-100,100,0]},Joined->True,PlotStyle->{Dashed, PointSize[Large],Black}
]]},Center]
, {{v,0},-.9,.9}]